3 cups water
2/3 cup plus 1 tablespoon sugar, divided
1 1/2 cups packed basil leaves
1/2 cup fresh lime juice, divided
1 ripe cantaloupe
1 ripe honeydew
1/4 cup mint leaves
Bring water and 2/3 cup sugar to a boil in a small saucepan, stirring until sugar has dissolved, then simmer 3 minutes. Transfer to a bowl and cool completely in an ice bath, stirring occasionally, about 10 minutes. Puree syrup with basil and 1/4 cup plus 2 tablespoon lime juice until smooth. Strain through a fine-mesh sieve into a 3-quart shallow nonreactive container, pressing on and then discarding solids. Freeze basil-lime syrup, stirring and crushing lumps with a fork every 30 minutes, until evenly frozen, about 2 hours. One hour before serving, cut melons into small wedges and peel. Combine in a large bowl. Whisk together remaining 2 tablespoon lime juice and 1 tablespoon sugar and pour over fruit. Let macerate, tossing occasionally, about 15 minutes. Scrape granita with a fork, crushing any lumps. Serve melon in shallow bowls, drizzled with melon liquid and topped with granita and mint.